Where, \(x(t_i), y(t_i), \theta (t_i) \)  are the position and heading of the vehicle at time, \(t_i\)  while \(u_1(t_i), u_2(t_i) \)  are the control inputs at some instant \(t_i\) . The terms, \(\kappa _{max}, \kappa _{min}\)  represent the maximum and minimum curvature bounds respectively. Thus, the inequalities in both the representations define the curvature bound constraints. The representation (REF ) is the classic non-holonomic model while (REF ) is obtained via feedback linearization [1]}. In (REF ), the curvature constraints are affine functions of control inputs. Thus, a trajectory optimization which uses (REF ) as the motion model is made difficult by the non-linear relationship between the position and the control variables. Conversely, in (REF ), the position variables are simply affine functions of the control inputs. Consequently, a trajectory optimization based on (REF ) is difficult due to the presence of non-linear curvature bounds. Thus, it can be seen that irrespective of which representation is used for the motion model of the non-holonomic vehicle, the trajectory optimization invariably takes the form of a difficult nonlinear and non-convex optimization problem. It is also interesting to note that if the curvature bounds are absent (e.g in case of differential drive robot), then using (REF ), the problem of connecting a pair of given positions/orientations in obstacle free space can be posed as a convex optimization problem. Further, if the obstacles (both static and dynamic) are approximated as circles, then the trajectory optimization with model (REF ) can be shown to have a difference of convex (DC) structure (e.g refer the derivations presented in [2]}, [3]}, [4]}). Specially designed sequential convex programming (SCP) routines exist for solving optimization problems with DC structure [5]}. However, inclusion of curvature constraints in (REF ) destroys the differnece of convex structure and consequently a more general non-linear optimization technique needs to be adopted [6]}.
